3.405 \(\int \frac{(a+c x^2+b x^4)^p}{(c+e x^2)^2} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\left (a+b x^4+c x^2\right )^p}{\left (c+e x^2\right )^2},x\right ) \]

[Out]

Defer[Int][(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2, x]

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Rubi [A]  time = 0.0107359, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2,x]

[Out]

Defer[Int][(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2, x]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx &=\int \frac{\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.266853, size = 0, normalized size = 0. \[ \int \frac{\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2,x]

[Out]

Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2, x]

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Maple [A]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{4}+c{x}^{2}+a \right ) ^{p}}{ \left ( e{x}^{2}+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+c*x^2+a)^p/(e*x^2+c)^2,x)

[Out]

int((b*x^4+c*x^2+a)^p/(e*x^2+c)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + c x^{2} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+c*x^2+a)^p/(e*x^2+c)^2,x, algorithm="maxima")

[Out]

integrate((b*x^4 + c*x^2 + a)^p/(e*x^2 + c)^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e^{2} x^{4} + 2 \, c e x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+c*x^2+a)^p/(e*x^2+c)^2,x, algorithm="fricas")

[Out]

integral((b*x^4 + c*x^2 + a)^p/(e^2*x^4 + 2*c*e*x^2 + c^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+c*x**2+a)**p/(e*x**2+c)**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + c x^{2} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+c*x^2+a)^p/(e*x^2+c)^2,x, algorithm="giac")

[Out]

integrate((b*x^4 + c*x^2 + a)^p/(e*x^2 + c)^2, x)